Explanation Definition used: Mutually exclusive events: The events A and B are termed as mutually exclusive if the intersection of the two events is the empty set. That is A ∩ B = ∅ where P ( A ∩ B ) = 0 . Also, for mutually exclusive events P ( A ∪ B ) = 1 . That is, A ∪ B = S . Formula used: Probability: If each simple event in sample space S is likely to occur as any other, then the probability of an arbitrary event E is given by, P ( E ) = Number of elements of E Number of elemenst of S = n ( E ) n ( S ) Independence condition: If A and B are any events in a sample space S , we say that A and B are independent if P ( A ∩ B ) = P ( A ) P ( B ) Description: Suppose A and B are mutually exclusive events, then these events cannot be independent, because for mutually exclusive events the probability of intersection for two events is 0. Hence the statement is false. The following example will disprove the given statement. Counter example: The possible outcomes when one coin is tossed are head ( H ) or tail ( T ). When a fair coin is tossed two times then generally the sample space for the experiment is, S = { H H , H T , T H , T T } . Define the event A as getting tail on the second toss and event B as getting head on the second toss. The probability for the outcome HH is, P ( H H ) = frequency of occurrence of H H total number of trials = 1 4 The probability for the outcome HT is, P ( H T ) = frequency of occurrence of H T total number of trials = 1 4 The probability for the outcome TH is, P ( T H ) = frequency of occurrence of T H total number of trials = 1 4 The probability for outcome TT is, P ( T T ) = frequency of occurrence of T T total number of trials = 1 4 The elements in the event A is, A = { H T , T T }

College Mathematics for Business, Economics, Life... (Instructor's)

14th Edition

ISBN: 9780134676104

Author: Barnett

Publisher: PEARSON

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Chapter 8.3, Problem 67E

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Whether the statement “If A and B are mutually exclusive, then A and B are independent” is true or not; explain if it is always true and give a counter example if it is not true.

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Chapter 8.3, Problem 66E

Chapter 8.3, Problem 68E

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College Mathematics for Business, Economics, Life... (Instructor's)

Ch. 8.1 - Repeat Example 1 for The outcome is a number...Ch. 8.1 - Prob. 2MP Ch. 8.1 - Prob. 3MP Ch. 8.1 - Suppose in Example 4 that after flipping the...Ch. 8.1 - Prob. 5MP Ch. 8.1 - Prob. 6MP Ch. 8.1 - Prob. 7MP Ch. 8.1 - Prob. 8MP Ch. 8.1 - Prob. 1ED Ch. 8.1 - Prob. 1E

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